Cauchy Convergence for Series

Question

We know that if $\sum a_{n}$ converges then $\lim_{n \rightarrow \infty} a_{n}=0$. Using this fact, I need to prove that if $\sum a_{n}$ converges, then for every $\epsilon > 0$ there is an integer K such that $$ \left|\sum_{n=k}^\infty a_{n}\right| < \epsilon \quad \textrm{if} \quad k \ge K, $$ that is, $$ \lim_{k \rightarrow \infty} \sum_{n=k}^\infty a_{n} = 0. $$ I know this is about the Cauchy Convergence Criterion, but I'm a bit confused with the premise and the form of the sum $|\sum_{n=k}^\infty a_{n}|$. This is usually expressed differently using n+p or something like that.

Answer 1

Let $s_k:= \sum_{n=1}^k a_{n}$ and $b_k:=\sum_{n=k}^\infty a_{n}$. Then we have

$b_k=\sum_{n=1}^\infty a_{n}-s_{k-1} \to\sum_{n=1}^\infty a_{n}-\sum_{n=1}^\infty a_{n}=0 $ as $k \to \infty$.